# Basic Material Properties

## Stress ( $$\sigma$$ ): Internal force per cross-sectional area

$$\sigma = \frac{F}{A}$$

## Strain ( $$\epsilon$$ ): Ratio of elongation to original

$$\epsilon = \frac{\Delta L}{L}$$

## Thermal Strain ( $$\epsilon _T$$ ), Thermal Stress ( $$\sigma _T$$ ):

$$\epsilon _T = \alpha \Delta L$$
$$\sigma _T = E \alpha \Delta L$$

Where $$\alpha$$ is the linear coefficient of thermal expansion.
If there are no external forces or restraints on the expansion, thermal stress is zero.

## Poisson's Ratio

Ratio of lateral strain to longitudinal strain.

$$v = - \frac{lateral}{longitudinal}$$

## Hooke's Law - Young's Modulus

$$E = \frac{\sigma}{\epsilon}$$

## Theory of Bending

$$\frac{M}{I} = \frac{\sigma}{y} = \frac{E}{R}$$

Where $$M$$ is the external moment acting on the beam, $$I$$ is the second moment of area, $$\sigma$$ is the stress of the material, $$y$$ is the distance from the neutral layer, $$E$$ is Young's Modulus, and $$R$$ is the radius of curvature.

$$I$$ for a rectangular cross section:

$$I = \frac{bd^3}{12}$$

Where $$b$$ is parallel to the neutral layer.

$$I$$ for a circular cross section:

$$I = \frac{\pi d^4}{64}$$

Where $$d$$ is the diameter of the cross section.

## Parallel Axis Theorem

$$I_{xx} = I_{gg} + Ah^2$$

Where $$I_{gg}$$ is the second moment of area passing through the centroid of body $$G$$, $$I_{xx}$$ is the second moment of area parallel to $$I_{gg}$$ at a distance $$h$$ away. $$A$$ is the area of body $$G$$.

## Average Shear Stress ( $$\tau$$ ):

$$\tau = \frac{shearforce}{forcearea} = \frac{P}{A}$$

Where $$P$$ is the internal shear force.

Assuming equilibrium conditions, average shear stress on the middle body is

$$\tau = \frac{\frac{P}{2}}{A}$$

## Shear Strain

$$\gamma \approx tan \gamma$$
$$G = \frac{\tau}{\gamma}$$

Where $$G$$ is the shear modulus.

## Shear Stress in Beams

$$\tau = \frac{VQ}{It}$$

Where $$V$$ is the internal shear force, $$Q$$ is the first moment of area of the half of the member's cross-section where $$t$$ is measured, $$I$$ is the second moment of area of the entire cross-section, $$t$$ is the width of the cross-section at the point where $$\tau$$ is to be determined.

## Torsion

$$\gamma = \frac{r \theta}{l}$$

Where $$r$$ is the radius of the beam, $$\theta$$ is the angle of distortion in radians, $$l$$ is the length of the beam.

$$\frac{ \tau }{r} = \frac{G \theta}{l}$$

The resultant torque $$T$$ for a thin-walled tube:

$$T = 2 \pi r^2 \tau t$$

Where $$t$$ is the thickness of the wall.

$$\frac{T}{J} = \frac{ \tau }{r} = \frac{G \theta}{l}$$

Where $$T$$ is the resultant torque, $$J$$ is the polar second moment of area, $$\tau$$ is the shear stress, $$r$$ is the radius of the beam, $$G$$ is the shear modulus, $$\theta$$ is the angle of distortion in radians, $$l$$ is the length of the beam.

The polar second moment of area for a shaft of circular cross-section is

$$J = \frac{\pi d^4}{32}$$

For a hollow shaft:

$$J = \frac{ \pi (d_2 ^4 - d_1 ^4 )}{32}$$

Where $$d_1$$ is the inner diameter, $$d_2$$ is the outer diameter.